Large Braids with Hair Particles
Well, I had play with the Children settings till actually got some reasonable result:



Here is my recipe for a nicer looking braid:


As you can see, mainly Random and Size properties made the strands get thicker, but it can also mess around with the shape at the end. So I suggest trim it with Length.
Use a high value for Clump on Effects Section and a low Clump on Kink Section.
If you get a fading effect like I was having on my question, try to change the Shape value to something like -0.9 on Cycles Hair Settings.




Suggestions are very welcome.